Problem: Find the least common multiple $(\text{LCM})$ of $y^4-16$ and $2y^4-8y^2$. You can give your answer in its factored form.
Answer: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $y^4-16$ can be factored as ${(y+2)(y-2)}{(y^2+4)}$ by using the difference of squares pattern twice. $2y^4-8y^2$ can be factored as ${(2)(y^2)}{(y+2)(y-2)}$ by factoring out a $2y^2$ and using the difference of squares pattern. We can see that: Both polynomials share the factors ${(y+2)(y-2)}$ Only the first polynomial has the factors ${(y^2+4)}$ Only the second polynomial has the factors ${(2)(y^2)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(y+2)(y-2)}{(y^2+4)}{(2)(y^2)}\\\\ &=2y^2(y+2)(y-2)(y^2+4)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $2y^2(y+2)(y-2)(y^2+4)$.